A259464 From higher-order arithmetic progressions.

75, 21875, 5512500, 1512630000, 484041600000, 184834742400000, 84715923600000000, 46534591303200000000, 30489464221856640000000, 23681690417572387200000000, 21660852835272876825600000000, 23175597788788462617600000000000, 28817200450516396946227200000000000

Offset: 0

Author

N. J. A. Sloane, Jun 30 2015

Keywords

• nonn

Comments

"3 over n!" on page 15 in the Dienger article is A087047; A_3 is A001303. - Georg Fischer, Dec 16 2024

Links

• Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]

Crossrefs

Cf. A001303, A087047, A259462, A259463.

Programs

Maple

rXI := proc(n, a, d)
        n*(n+1)*(n+2)/6*a+(n+2)*(n+1)*n*(n-1)/24*d;
end proc:
A259464 := proc(n)
        mul(rXI(i, a, d), i=1..n+3) ;
        coeftayl(%, d=0, 3) ;
        coeftayl(%, a=0, n) ;
end proc:
seq(A259464(n), n=1..25) ; # R. J. Mathar, Jul 15 2015

Mathematica

rXI[n_, a_, d_] := (n(n+1)(n+2)/6)*a+((n+2)(n+1)n(n-1)/24)*d;
A259464[n_] :=
   Product[rXI[i, a, d], {i, 1, n+4}]//
   SeriesCoefficient[#, {d, 0, 3}]&//
   SeriesCoefficient[#, {a, 0, n+1}]&;
Table[A259464[n], {n, 0, 12}] (* Jean-François Alcover, Apr 26 2023, after R. J. Mathar *)

Formula

D-finite with recurrence: -6*n*(n+2)*a(n) +(n+6)*(n+5)*(n+4)^3*a(n-1)=0. - R. J. Mathar, Jul 15 2015

a(n) = 2^(-n-5)*3^(-n-4)*(n+4)!*(n+5)!*(n+6)!*(n+4)^2*(n+3)^2*(n+2)*(n+1)/3072. - Georg Fischer, Dec 16 2024